\(\int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx\) [126]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \]

[Out]

-3/2*e^2*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^4-3/2*(-e^2*x^2+d^2)^(1/2)/d^3/x^2+2*e*(-e^2*x^2+d^2)^(1/2)/d^4/x+(
-e^2*x^2+d^2)^(1/2)/d^2/x^2/(e*x+d)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {871, 849, 821, 272, 65, 214} \[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2} \]

[In]

Int[1/(x^3*(d + e*x)*Sqrt[d^2 - e^2*x^2]),x]

[Out]

(-3*Sqrt[d^2 - e^2*x^2])/(2*d^3*x^2) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^4*x) + Sqrt[d^2 - e^2*x^2]/(d^2*x^2*(d + e
*x)) - (3*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^4)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 871

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[d*(f + g*x)^
(n + 1)*((a + c*x^2)^(p + 1)/(2*a*p*(e*f - d*g)*(d + e*x))), x] + Dist[1/(p*(2*c*d)*(e*f - d*g)), Int[(f + g*x
)^n*(a + c*x^2)^p*(c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x], x] /; FreeQ[{a, c, d, e
, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[n, 0] && ILtQ[n + 2*p, 0] &
&  !IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {\int \frac {-3 d e^2+2 e^3 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2} \\ & = -\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {\int \frac {-4 d^2 e^3+3 d e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^4 e^2} \\ & = -\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {\left (3 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^3} \\ & = -\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^3} \\ & = -\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{2 d^3} \\ & = -\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (-d^2+d e x+4 e^2 x^2\right )}{x^2 (d+e x)}-3 \sqrt {d^2} e^2 \log (x)+3 \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{2 d^5} \]

[In]

Integrate[1/(x^3*(d + e*x)*Sqrt[d^2 - e^2*x^2]),x]

[Out]

((d*Sqrt[d^2 - e^2*x^2]*(-d^2 + d*e*x + 4*e^2*x^2))/(x^2*(d + e*x)) - 3*Sqrt[d^2]*e^2*Log[x] + 3*Sqrt[d^2]*e^2
*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(2*d^5)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-2 e x +d \right )}{2 d^{4} x^{2}}-\frac {3 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{3} \sqrt {d^{2}}}+\frac {e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{4} \left (x +\frac {d}{e}\right )}\)	\(117\)
default	\(\frac {-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}}{d}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{3} \sqrt {d^{2}}}+\frac {e \sqrt {-e^{2} x^{2}+d^{2}}}{d^{4} x}+\frac {e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{4} \left (x +\frac {d}{e}\right )}\)	\(183\)

[In]

int(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-e^2*x^2+d^2)^(1/2)*(-2*e*x+d)/d^4/x^2-3/2*e^2/d^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1
/2))/x)+e/d^4/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\frac {2 \, e^{3} x^{3} + 2 \, d e^{2} x^{2} + 3 \, {\left (e^{3} x^{3} + d e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (4 \, e^{2} x^{2} + d e x - d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, {\left (d^{4} e x^{3} + d^{5} x^{2}\right )}} \]

[In]

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*(2*e^3*x^3 + 2*d*e^2*x^2 + 3*(e^3*x^3 + d*e^2*x^2)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (4*e^2*x^2 + d*e*x
- d^2)*sqrt(-e^2*x^2 + d^2))/(d^4*e*x^3 + d^5*x^2)

Sympy [F]

\[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{x^{3} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]

[In]

integrate(1/x**3/(e*x+d)/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(x**3*sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)), x)

Maxima [F]

\[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )} x^{3}} \,d x } \]

[In]

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^3), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (101) = 202\).

Time = 0.29 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.27 \[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\frac {{\left (e^{3} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e}{x} - \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e x^{2}}\right )} e^{4} x^{2}}{8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} - \frac {3 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{4} {\left | e \right |}} + \frac {\frac {4 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} e {\left | e \right |}}{x} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} {\left | e \right |}}{e x^{2}}}{8 \, d^{8} e^{2}} \]

[In]

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

1/8*(e^3 - 3*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e/x - 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e*x^2))*e^4*x
^2/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^4*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)*abs(e)) - 3/2*
e^3*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^4*abs(e)) + 1/8*(4*(d*e + sqrt(-e^2*x
^2 + d^2)*abs(e))*d^4*e*abs(e)/x - (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^4*abs(e)/(e*x^2))/(d^8*e^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{x^3\,\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \]

[In]

int(1/(x^3*(d^2 - e^2*x^2)^(1/2)*(d + e*x)),x)

[Out]

int(1/(x^3*(d^2 - e^2*x^2)^(1/2)*(d + e*x)), x)